Sums of Binomial Coefficients
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Due Mar 2, 2:59 AM EST
What is the sum of all the elements in the sixth rows of Pascal's triangle?

Exactly! Indeed, the sum of binomial coefficients in the i-th row is equal to 2i
What is the sum of all the elements from the first six rows of Pascal's triangle?

Exactly! It is 1+2+4+8+16+32=21+22+23+24+25=26−1=63.
What is the value of the sum (35)+(45)+(55)?

Exactly! We can compute the sum explicitly, but we can also observe that for all i we have (i5)=(5−i5). So, the sum of these three coefficients is exactly half of the sum of all coefficients in the sixth row.